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Basic Experimental Design Larry V. Hedges Northwestern University Prepared for the IES Summer Research Training Institute July 8, 2008

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What is Experimental Design? Experimental design includes both Strategies for organizing data collection Data analysis procedures matched to those data collection strategies Classical treatments of design stress analysis procedures based on the analysis of variance (ANOVA) Other analysis procedure such as those based on hierarchical linear models or analysis of aggregates (e.g., class or school means) are also appropriate

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Why Do We Need Experimental Design? Because of variability We wouldn’t need a science of experimental design if If all units (students, teachers, & schools) were identical and If all units responded identically to treatments We need experimental design to control variability so that treatment effects can be identified

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A Little History The idea of controlling variability through design has a long history In 1747 Sir James Lind’s studies of scurvy Their cases were as similar as I could have them. They all in general had putrid gums, spots and lassitude, with weakness of their knees. They lay together on one place … and had one diet common to all (Lind, 1753, p. 149) Lind then assigned six different treatments to groups of patients

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A Little History The idea of random assignment was not obvious and took time to catch on In 1648 von Helmont carried out one randomization in a trial of bloodletting for fevers In 1904 Karl Pearson suggested matching and alternation in typhoid trials Amberson, et al. (1931) carried out a trial with one randomization In 1937 Sir Bradford Hill advocated alternation of patients in trials rather than randomization Diehl, et al. (1938) carried out a trial that is sometimes referred to as randomized, but it actually used alternation

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A Little History The first modern randomized clinical trial in medicine is usually considered to be the trial of streptomycin for treating tuberculosis It was conducted by the British Medical Research Council in 1946 and reported in 1948

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A Little History Experiments have been used longer in the behavioral sciences (e.g., psychophysics: Pierce and Jastrow, 1885) Experiments conducted in laboratory settings were widely used in educational psychology (e.g., McCall, 1923) Thorndike (early 1900’s) Lindquist (1953) Gage field experiments on teaching (1978 – 1984)

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A Little History Studies in crop variation I – VI (1921 – 1929) In 1919 a statistician named Fisher was hired at Rothamsted agricultural station They had a lot of observational data on crop yields and hoped a statistician could analyze it to find effects of various treatments All he had to do was sort out the effects of confounding variables

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Studies in Crop Variation I (1921) Fisher does regression analyses—lots of them—to study (and get rid of) the effects of confounders soil fertility gradients drainage effects of rainfall effects of temperature and weather, etc. Fisher does qualitative work to sort out anomalies Conclusion The effects of confounders are typically larger than those of the systematic effects we want to study

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Studies in Crop Variation II (1923) Fisher invents Basic principles of experimental design Control of variation by randomization Analysis of variance

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Studies in Crop Variation IV and VI Studies in Crop variation IV (1927) Fisher invents analysis of covariance to combine statistical control and control by randomization Studies in crop variation VI (1929) Fisher refines the theory of experimental design, introducing most other key concepts known today

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Our Hero in 1929

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Principles of Experimental Design Experimental design controls background variability so that systematic effects of treatments can be observed Three basic principles 1.Control by matching 2.Control by randomization 3.Control by statistical adjustment Their importance is in that order

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Control by Matching Known sources of variation may be eliminated by matching Eliminating genetic variation Compare animals from the same litter of mice Eliminating district or school effects Compare students within districts or schools However matching is limited matching is only possible on observable characteristics perfect matching is not always possible matching inherently limits generalizability by removing (possibly desired) variation

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Control by Matching Matching ensures that groups compared are alike on specific known and observable characteristics (in principle, everything we have thought of) Wouldn’t it be great if there were a method of making groups alike on not only everything we have thought of, but everything we didn’t think of too? There is such a method

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Control by Randomization Matching controls for the effects of variation due to specific observable characteristics Randomization controls for the effects all (observable or non-observable, known or unknown) characteristics Randomization makes groups equivalent (on average) on all variables (known and unknown, observable or not) Randomization also gives us a way to assess whether differences after treatment are larger than would be expected due to chance.

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Control by Randomization Random assignment is not assignment with no particular rule. It is a purposeful process Assignment is made at random. This does not mean that the experimenter writes down the names of the varieties in any order that occurs to him, but that he carries out a physical experimental process of randomization, using means which shall ensure that each variety will have an equal chance of being tested on any particular plot of ground (Fisher, 1935, p. 51)

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Control by Randomization Random assignment of schools or classrooms is not assignment with no particular rule. It is a purposeful process Assignment of schools to treatments is made at random. This does not mean that the experimenter assigns schools to treatments in any order that occurs to her, but that she carries out a physical experimental process of randomization, using means which shall ensure that each treatment will have an equal chance of being tested in any particular school (Hedges, 2007)

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Control by Statistical Adjustment Control by statistical adjustment is a form of pseudo- matching It uses statistical relations to simulate matching Statistical control is important for increasing precision but should not be relied upon to control biases that may exist prior to assignment Statistical control is the weakest of the three experimental design principles because its validity depends on knowing a statistical model for responses

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Using Principles of Experimental Design You have to know a lot (be smart) to use matching and statistical control effectively You do not have to be smart to use randomization effectively But Where all are possible, randomization is not as efficient (requires larger sample sizes for the same power) as matching or statistical control

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Basic Ideas of Design: Independent Variables (Factors) The values of independent variables are called levels Some independent variables can be manipulated, others can’t Treatments are independent variables that can be manipulated Blocks and covariates are independent variables that cannot be manipulated These concepts are simple, but are often confused Remember: You can randomly assign treatment levels but not blocks

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Basic Ideas of Design (Crossing) Relations between independent variables Factors (treatments or blocks) are crossed if every level of one factor occurs with every level of another factor Example The Tennessee class size experiment assigned students to one of three class size conditions. All three treatment conditions occurred within each of the participating schools Thus treatment was crossed with schools

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Basic Ideas of Design (Nesting) Factor B is nested in factor A if every level of factor B occurs within only one level of factor A Example The Tennessee class size experiment actually assigned classrooms to one of three class size conditions. Each classroom occurred in only one treatment condition Thus classrooms were nested within treatments (But treatment was crossed with schools)

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Where Do These Terms Come From? (Nesting) An agricultural experiment where blocks are literally blocks or plots of land Here each block is literally nested within a treatment condition Blocks 12…n T1T2…T1

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Where Do These Terms Come From? (Crossing) An agricultural experiment Blocks were literally blocks of land and plots of land within blocks were assigned different treatments Blocks 12…n T1T2 … T1 T2T1T2

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Where Do These Terms Come From? (Crossing) Blocks were literally blocks of land and plots of land within blocks were assigned different treatments. Here treatment literally crosses the blocks Blocks 12…n T1T2 … T1 T2T1T2

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Where Do These Terms Come From? (Crossing) The experiment is often depicted like this. What is wrong with this as a field layout? Consider possible sources of bias Blocks 12…n Treatment 1 … Treatment 2

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Think About These Designs A study assigns a reading treatment (or control) to children in 20 schools. Each child is classified into one of three groups with different risk of reading failure. A study assigns T or C to 20 teachers. The teachers are in five schools, and each teacher teaches 4 science classes Two schools in each district are picked to participate. Each school has two grade 4 teachers. One of them is assigned to T, the other to C.

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Three Basic Designs The completely randomized design Treatments are assigned to individuals The randomized block design Treatments are assigned to individuals within blocks (This is sometimes called the matched design, because individuals are matched within blocks) The hierarchical design Treatments are assigned to blocks, the same treatment is assigned to all individuals in the block

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The Completely Randomized Design Individuals are randomly assigned to one of two treatments TreatmentControl Individual 1 Individual 2 …… Individual n T Individual n C

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The Randomized Block Design Block 1…Block m Treatment 1 Individual 1 … …… … …… Individual n 1 Individual n m Treatment 2 Individual n 1 +1Individual n m + 1 …… Individual 2n 1 Individual 2n m

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The Hierarchical Design TreatmentControl Block 1Block mBlock m+1Block 2m Individual 1 … … Individual 2 ………… Individual n 1 Individual n m Individual n m+1 Individual n 2m

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Randomization Procedures Randomization has to be done as an explicit process devised by the experimenter Haphazard is not the same as random Unknown assignment is not the same as random “Essentially random” is technically meaningless Alternation is not random, even if you alternate from a random start This is why R.A. Fisher was so explicit about randomization processes

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Randomization Procedures R.A. Fisher on how to randomize an experiment with small sample size and 5 treatments A satisfactory method is to use a pack of cards numbered from 1 to 100, and to arrange them in random order by repeated shuffling. The varieties [treatments] are numbered from 1 to 5, and any card such as the number 33, for example is deemed to correspond to variety [treatment] number 3, because on dividing by 5 this number is found as the remainder. (Fisher, 1935, p.51)

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Randomization Procedures You may want to use a table of random numbers, but be sure to pick an arbitrary start point! Beware random number generators—they typically depend on seed values, be sure to vary the seed value (if they do not do it automatically) Otherwise you can reliably generate the same sequence of random numbers every time It is no different that starting in the same place in a table of random numbers

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Randomization Procedures Completely Randomized Design (2 treatments, 2n individuals) Make a list of all individuals For each individual, pick a random number from 1 to 2 (odd or even) Assign the individual to treatment 1 if even, 2 if odd When one treatment is assigned n individuals, stop assigning more individuals to that treatment

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Randomization Procedures Completely Randomized Design (2pn individuals, p treatments) Make a list of all individuals For each individual, pick a random number from 1 to p One way to do this is to get a random number of any size, divide by p, the remainder R is between 0 and (p – 1), so add 1 to the remainder to get R + 1 Assign the individual to treatment R + 1 Stop assigning individuals to any treatment after it gets n individuals

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Randomization Procedures Randomized Block Design with 2 Treatments (m blocks per treatment, 2n individuals per block) Make a list of all individuals in the first block For each individual, pick a random number from 1 to 2 (odd or even) Assign the individual to treatment 1 if even, 2 if odd Stop assigning a treatment it is assigned n individuals in the block Repeat the same process with every block

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Randomization Procedures Randomized Block Design with p Treatments (m blocks per treatment, pn individuals per block) Make a list of all individuals in the first block For each individual, pick a random number from 1 to p Assign the individual to treatment p Stop assigning a treatment it is assigned n individuals in the block Repeat the same process with every block

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Randomization Procedures Hierarchical Design with 2 Treatments (m blocks per treatment, n individuals per block) Make a list of all blocks For each block, pick a random number from 1 to 2 Assign the block to treatment 1 if even, treatment 2 if odd Stop assigning a treatment after it is assigned m blocks Every individual in a block is assigned to the same treatment

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Randomization Procedures Hierarchical Design with p Treatments (m blocks per treatment, n individuals per block) Make a list of all blocks For each block, pick a random number from 1 to p Assign the block to treatment corresponding to the number Stop assigning a treatment after it is assigned m blocks Every individual in a block is assigned to the same treatment

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Sampling Models

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Sampling Models in Educational Research Sampling models are often ignored in educational research But Sampling is where the randomness comes from in social research Sampling therefore has profound consequences for statistical analysis and research designs

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Sampling Models in Educational Research Simple random samples are rare in field research Educational populations are hierarchically nested: Students in classrooms in schools Schools in districts in states We usually exploit the population structure to sample students by first sampling schools Even then, most samples are not probability samples, but they are intended to be representative (of some population)

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Sampling Models in Educational Research Survey research calls this strategy multistage (multilevel) clustered sampling We often sample clusters (schools) first then individuals within clusters (students within schools) This is a two-stage (two-level) cluster sample We might sample schools, then classrooms, then students This is a three-stage (three-level) cluster sample

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Precision of Estimates Depends on the Sampling Model Suppose the total population variance is σ T 2 and ICC is ρ Consider two samples of size N = mn A simple random sample or stratified sample The variance of the mean is σ T 2 /mn A clustered sample of n students from each of m schools The variance of the mean is (σ T 2 /mn)[1 + (n – 1)ρ] The inflation factor [1 + (n – 1)ρ] is called the design effect

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Precision of Estimates Depends on the Sampling Model Suppose the population variance is σ T 2 School level ICC is ρ S, class level ICC is ρ C Consider two samples of size N = mpn A simple random sample or stratified sample The variance of the mean is σ T 2 /mpn A clustered sample of n students from p classes in m schools The variance is (σ T 2 /mpn)[1 + (pn – 1)ρ S + (n – 1)ρ C ] The three level design effect is [1 + (pn – 1)ρ S + (n – 1)ρ C ]

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Precision of Estimates Depends on the Sampling Model Treatment effects in experiments and quasi- experiments are mean differences Therefore precision of treatment effects and statistical power will depend on the sampling model

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Sampling Models in Educational Research The fact that the population is structured does not mean the sample is must be a clustered sample Whether it is a clustered sample depends on: How the sample is drawn (e.g., are schools sampled first then individuals randomly within schools) What the inferential population is (e.g., is the inference to these schools studied or a larger population of schools)

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Sampling Models in Educational Research A necessary condition for a clustered sample is that it is drawn in stages using population subdivisions schools then students within schools schools then classrooms then students However, if all subdivisions in a population are present in the sample, the sample is not clustered, but stratified Stratification has different implications than clustering Whether there is stratification or clustering depends on the definition of the population to which we draw inferences (the inferential population)

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Sampling Models in Educational Research The clustered/stratified distinction matters because it influences the precision of statistics estimated from the sample If all population subdivisions are included in the every sample, there is no sampling (or exhaustive sampling) of subdivisions therefore differences between subdivisions add no uncertainty to estimates If only some population subdivisions are included in the sample, it matters which ones you happen to sample thus differences between subdivisions add to uncertainty

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Inferential Population and Inference Models The inferential population or inference model has implications for analysis and therefore for the design of experiments Do we make inferences to the schools in this sample or to a larger population of schools? Inferences to the schools or classes in the sample are called conditional inferences Inferences to a larger population of schools or classes are called unconditional inferences

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Inferential Population and Inference Models Note that the inferences (what we are estimating) are different in conditional versus unconditional inference models In a conditional inference, we are estimating the mean (or treatment effect) in the observed schools In unconditional inference we are estimating the mean (or treatment effect) in the population of schools from which the observed schools are sampled We are still estimating a mean (or a treatment effect) but they are different parameters with different uncertainties

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Fixed and Random Effects When the levels of a factor (e.g., particular blocks included) in a study are sampled and the inference model is unconditional, that factor is called random and its effects are called random effects When the levels of a factor (e.g., particular blocks included) in a study constitute the entire inference population and the inference model is conditional, that factor is called fixed and its effects are called fixed effects

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Applications to Experimental Design We will look in detail at the two most widely used experimental designs in education Randomized blocks designs Hierarchical designs

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Experimental Designs For each design we will look at Structural Model for data (and what it means) Two inference models –What does ‘treatment effect’ mean in principle –What is the estimate of treatment effect –How do we deal with context effects Two statistical analysis procedures –How do we estimate and test treatment effects –How do we estimate and test context effects –What is the sensitivity of the tests

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The Randomized Block Design The population (the sampling frame) We wish to compare two treatments We assign treatments within schools Many schools with 2 n students in each Assign n students to each treatment in each school

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The Randomized Block Design The experiment Compare two treatments in an experiment We assign treatments within schools With m schools with 2 n students in each Assign n students to each treatment in each school

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The Randomized Block Design Diagram of the design Schools Treatment12… m 1 … 2 …

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The Randomized Block Design School 1 Schools Treatment12… m 1 … 2 …

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The Conceptual Model The statistical model for the observation on the k th person in the j th school in the i th treatment is Y ijk = μ +α i + β j + αβ ij + ε ijk where μ is the grand mean, α i is the average effect of being in treatment i, β j is the average effect of being in school j, αβ ij is the difference between the average effect of treatment i and the effect of that treatment in school j, ε ijk is a residual

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Effect of Context Context Effect

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Two-level Randomized Block Design With No Covariates (HLM Notation) Level 1 (individual level) Y ijk = β 0j + β 1j T ijk + ε ijk ε ~ N(0, σ W 2 ) Level 2 (school Level) β 0j = π 00 + ξ 0j ξ 0j ~ N(0, σ S 2 ) β 1j = π 10 + ξ 1j ξ 1j ~ N(0, σ T x S 2 ) If we code the treatment T ijk = ½ or - ½, then the parameters are identical to those in standard ANOVA

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Effects and Estimates The population mean of treatment 1 in school j is α 1 + αβ 1j The population mean of treatment 2 in school j is α 2 + αβ 2j The estimate of the mean of treatment 1 in school j is α 1 + αβ 1j + ε 1j ● The estimate of the mean of treatment 2 in school j is α 2 + αβ 2j + ε 2j ●

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Effects and Estimates The comparative treatment effect in any given school j is ( α 1 – α 2 ) + ( αβ 1j – αβ 2j ) The estimate of comparative treatment effect in school j is (α 1 – α 2 ) + ( αβ 1j – αβ 2j ) + ( ε 1j ● – ε 2j ● ) The mean treatment effect in the experiment is (α 1 – α 2 ) + ( αβ 1 ● – αβ 2 ● ) The estimate of the mean treatment effect in the experiment is (α 1 – α 2 ) + ( αβ 1 ● – αβ 2 ● ) + ( ε 1 ●● – ε 2 ●● )

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Inference Models Two different kinds of inferences about effects Unconditional Inference (Schools Random) Inference to the whole universe of schools (requires a representative sample of schools) Conditional Inference (Schools Fixed) Inference to the schools in the experiment (no sampling requirement on schools)

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Statistical Analysis Procedures Two kinds of statistical analysis procedures Mixed Effects Procedures (Schools Random) Treat schools in the experiment as a sample from a population of schools (only strictly correct if schools are a sample) Fixed Effects Procedures (Schools Fixed) Treat schools in the experiment as a population

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Unconditional Inference (Schools Random) The estimate of the mean treatment effect in the experiment is (α 1 – α 2 ) + ( αβ 1 ● – αβ 2 ● ) + ( ε 1 ●● – ε 2 ●● ) The average treatment effect we want to estimate is (α 1 – α 2 ) The term ( ε 1 ●● – ε 2 ●● ) depends on the students in the schools in the sample The term ( αβ 1 ● – αβ 2 ● ) depends on the schools in sample Both ( ε 1 ●● – ε 2 ●● ) and ( αβ 1 ● – αβ 2 ● ) are random and average to 0 across students and schools, respectively

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Conditional Inference (Schools Fixed) The estimate of the mean treatment effect in the experiment is still (α 1 – α 2 ) + ( αβ 1 ● – αβ 2 ● ) + ( ε 1 ●● – ε 2 ●● ) Now the average treatment effect we want to estimate is (α 1 + αβ 1● ) – (α 2 + αβ 2● ) = (α 1 – α 2 ) + (αβ 1● – αβ 2● ) The term ( ε 1 ●● – ε 2 ●● ) depends on the students in the schools in the sample The term ( αβ 1 ● – αβ 2 ● ) depends on the schools in sample, but the treatment effect in the sample of schools is the effect we want to estimate

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Expected Mean Squares Randomized Block Design (Two Levels, Schools Random) Source df E{MS} Treatment (T)1 σ W 2 + nσ T x S 2 + nmΣα i 2 Schools (S) m – 1 σ W 2 + 2nσ S 2 T X S m – 1 σ W 2 + nσ T x S 2 Within Cells 2 m ( n – 1) σW2σW2

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Mixed Effects Procedures (Schools Random) The test for treatment effects has H 0 : (α 1 – α 2 ) = 0 Estimated mean treatment effect in the experiment is (α 1 – α 2 ) + ( αβ 1 ● – αβ 2 ● ) + ( ε 1 ●● – ε 2 ●● ) The variance of the estimated treatment effect is 2[σ W 2 + nσ T x S 2 ] /mn = 2[1 + (n ω S – 1)ρ]σ 2 /mn Here ω S = σ T x S 2 /σ S 2 and ρ = σ S 2 /(σ S 2 + σ W 2 ) = σ S 2 /σ 2

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Mixed Effects Procedures The test for treatment effects: F T = MS T /MS T x S with (m – 1) df The test for context effects (treatment by schools interaction) is F TxS = MS T x S /MS WS with 2m(n – 1) df Power is determined by the operational effect size where ω S = σ T x S 2 /σ S 2 and ρ = σ S 2 /(σ S 2 + σ W 2 ) = σ S 2 /σ 2

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Expected Mean Squares Randomized Block Design (Two Levels, Schools Fixed) Source df E{MS} Treatment (T)1 σ W 2 + nmΣα i 2 Schools (S) m – 1 σ W 2 + 2nΣβ i 2 /(m – 1) S X T m – 1 σ W 2 + nΣΣαβ ij 2 /(m – 1) Within Cells 2 m ( n – 1) σW2σW2

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Fixed Effects Procedures The test for treatment effects has H 0 : (α 1 – α 2 ) + (αβ 1● – αβ 2● ) = 0 Estimated mean treatment effect in the experiment is (α 1 – α 2 ) + (αβ 1 ● – αβ 2 ● ) + ( ε 1 ●● – ε 2 ●● ) The variance of the estimated treatment effect is 2σ W 2 /mn

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Fixed Effects Procedures The test for treatment effects: F T = MS T /MS WS with m(n – 1) df The test for context effects (treatment by schools interaction) is F C = MS T x S /MS WS with 2 m(n – 1) df Power is determined by the operational effect size with m(n – 1) df

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Comparing Fixed and Mixed Effects Statistical Procedures (Randomized Block Design) FixedMixed Inference ModelConditionalUnconditional Estimand (α 1 – α 2 ) + (αβ 1● – αβ 2● )(α 1 – α 2 ) Contaminating Factors (ε 1●● – ε 2●● )(αβ 1● – αβ 2● ) + (ε 1●● – ε 2●● ) Operational Effect Size df 2m(n – 1)(m – 1) Powerhigherlower

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Comparing Fixed and Mixed Effects Procedures (Randomized Block Design) Conditional and unconditional inference models estimate different treatment effects have different contaminating factors that add uncertainty Mixed procedures are good for unconditional inference The fixed procedures are good for conditional inference The fixed procedures have higher power

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The Hierarchical Design The universe (the sampling frame) We wish to compare two treatments We assign treatments to whole schools Many schools with n students in each Assign all students in each school to the same treatment

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The Hierarchical Design The experiment We wish to compare two treatments We assign treatments to whole schools Assign 2 m schools with n students in each Assign all students in each school to the same treatment

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The Hierarchical Design Diagram of the experiment Schools Treatment12… m m +1m +2… 2m2m 1 2

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The Hierarchical Design Treatment 1 schools Schools Treatment12… m m +1m + 2…2 m 1 2

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The Hierarchical Design Treatment 2 schools Schools Treatment12… m m + 1m + 2…2 m 1 2

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The Conceptual Model The statistical model for the observation on the k th person in the j th school in the i th treatment is Y ijk = μ + α i + β i + αβ ij + ε jk(i) = μ + α i + β j(i) + ε jk(i) μ is the grand mean, α i is the average effect of being in treatment i, β j is the average effect if being in school j, αβ ij is the difference between the average effect of treatment i and the effect of that treatment in school j, ε ijk is a residual Or β j(i) = β i + αβ ij is a term for the combined effect of schools within treatments

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The Conceptual Model The statistical model for the observation on the k th person in the j th school in the i th treatment is Y ijk = μ + α i + β i + αβ ij + ε jk(i) = μ + α i + β j(i) + ε jk(i) μ is the grand mean, α i is the average effect of being in treatment i, β j is the average effect if being in school j, αβ ij is the difference between the average effect of treatment i and the effect of that treatment in school j, ε ijk is a residual or β j(i) = β i + αβ ij is a term for the combined effect of schools within treatments Context Effects

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Two-level Hierarchical Design With No Covariates (HLM Notation) Level 1 (individual level) Y ijk = β 0j + ε ijk ε ~ N(0, σ W 2 ) Level 2 (school Level) γ 0j = π 00 + π 01 T j + ξ 0j ξ ~ N(0, σ S 2 ) If we code the treatment T j = ½ or - ½, then π 00 = μ, π 01 = α 1, ξ 0j = β j(i) The intraclass correlation is ρ = σ S 2 /(σ S 2 + σ W 2 ) = σ S 2 /σ 2

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Effects and Estimates The comparative treatment effect in any given school j is still (α 1 – α 2 ) + ( αβ 1j – αβ 2j ) But we cannot estimate the treatment effect in a single school because each school gets only one treatment The mean treatment effect in the experiment is (α 1 – α 2 ) + (β ● (1) – β ● (2) ) = (α 1 – α 2 ) +(β 1 ● – β 2 ● )+ ( αβ 1 ● – αβ 2 ● ) The estimate of the mean treatment effect in the experiment is (α 1 – α 2 ) + (β ● (1) – β ● (2) ) + ( ε 1 ●● – ε 2 ●● )

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Inference Models Two different kinds of inferences about effects (as in the randomized block design) Unconditional Inference (schools random) Inference to the whole universe of schools (requires a representative sample of schools) Conditional Inference (schools fixed) Inference to the schools in the experiment (no sampling requirement on schools)

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Unconditional Inference (Schools Random) The average treatment effect we want to estimate is (α 1 – α 2 ) The term ( ε 1 ●● – ε 2 ●● ) depends on the students in the schools in the sample The term (β ● (1) – β ● (2) ) depends on the schools in sample Both ( ε 1 ●● – ε 2 ●● ) and (β ● (1) – β ● (2) ) are random and average to 0 across students and schools, respectively

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Conditional Inference (Schools Fixed) The average treatment effect we want to (can) estimate is (α 1 + β ●(1) ) – (α 2 + β ●(2) ) = (α 1 – α 2 ) + (β ●(1) – β ●(2) ) = (α 1 – α 2 ) + (β 1 ● – β 2 ● )+ (αβ 1 ● – αβ 2 ● ) The term (β ● (1) – β ● (2) ) depends on the schools in sample, but we want to estimate the effect of treatment in the schools in the sample Note that this treatment effect is not quite the same as in the randomized block design, where we estimate (α 1 – α 2 ) + (αβ 1 ● – αβ 2 ● )

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Statistical Analysis Procedures Two kinds of statistical analysis procedures (as in the randomized block design) Mixed Effects Procedures Treat schools in the experiment as a sample from a universe Fixed Effects Procedures Treat schools in the experiment as a universe

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Expected Mean Squares Hierarchical Design (Two Levels, Schools Random) Source df E{MS} Treatment (T)1 σ W 2 + nσ S 2 + nmΣα i 2 Schools (S)2( m – 1) σ W 2 + nσ S 2 Within Schools2 m ( n – 1) σW2σW2

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Mixed Effects Procedures (Schools Random) The test for treatment effects has H 0 : (α 1 – α 2 ) = 0 Estimated mean treatment effect in the experiment is (α 1 – α 2 ) + (β ● (1) – β ● (2) ) + ( ε 1 ●● – ε 2 ●● ) The variance of the estimated treatment effect is 2[σ W 2 + nσ S 2 ] /mn = 2[1 + (n – 1)ρ]σ 2 /mn where ρ = σ S 2 /(σ S 2 + σ W 2 ) = σ S 2 /σ 2

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Mixed Effects Procedures (Schools Random) The test for treatment effects: F T = MS T /MS BS with (m – 2) df There is no omnibus test for context effects Power is determined by the operational effect size where ρ = σ S 2 /(σ S 2 + σ W 2 ) = σ S 2 /σ 2

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Expected Mean Squares Hierarchical Design (Two Levels, Schools Fixed) Source df E{MS} Treatment (T)1 σ W 2 + nmΣ(α i + β ● (i) ) 2 Schools (S) m – 1 σ W 2 + nΣΣβ j (i) 2 /2(m – 1) Within Schools2 m ( n – 1) σW2σW2

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Mixed Effects Procedures (Schools Fixed) The test for treatment effects has H 0 : (α 1 – α 2 ) + (β ●(1) – β ●(2) ) = 0 Note that the school effects are confounded with treatment effects Estimated mean treatment effect in the experiment is (α 1 – α 2 ) + (β ● (1) – β ● (2) ) + ( ε 1 ●● – ε 2 ●● ) The variance of the estimated treatment effect is 2σ W 2 /mn

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Mixed Effects Procedures (Schools Fixed) The test for treatment effects: F T = MS T /MS WS with m(n – 1) df There is no omnibus test for context effects, because each school gets only one treatment Power is determined by the operational effect size and m(n – 1) df

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Comparing Fixed and Mixed Effects Procedures (Hierarchical Design) FixedMixed Inference Model ConditionalUnconditional Estimand (α 1 – α 2 ) + (β ●(1) – β ●(2) )(α 1 – α 2 ) Contaminating Factors (ε 1●● – ε 2●● )(β ●(1) – β ●(2) ) + (ε 1●● – ε 2●● ) Effect Size df m(n – 1)(m – 2) Powerhigherlower

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Comparing Fixed and Mixed Effects Statistical Procedures (Hierarchical Design) Conditional and unconditional inference models estimate different treatment effects have different contaminating factors that add uncertainty Mixed procedures are good for unconditional inference The fixed procedures are not generally recommended The fixed procedures have higher power

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Comparing Hierarchical Designs to Randomized Block Designs Randomized block designs usually have higher power, but assignment of different treatments within schools or classes may be practically difficult politically infeasible theoretically impossible It may be methodologically unwise because of potential for Contamination or diffusion of treatments compensatory rivalry or demoralization

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Applications to Experimental Design We will address the two most widely used experimental designs in education Randomized blocks designs with 2 levels Randomized blocks designs with 3 levels Hierarchical designs with 2 levels Hierarchical designs with 3 levels We also examine the effect of covariates Hereafter, we generally take schools to be random

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Complications Which matchings do we have to take into account in design (e.g., schools, districts, regions, states, regions of the country, country)? Ignore some, control for effects of others as fixed blocking factors Justify this as part of the population definition For example, we define the inference population as these five districts within these two states But, doing so obviously constrains generalizability

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Precision of the Estimated Treatment Effect Precision is the standard error of the estimated treatment effect Precision in simple (simple random sample) designs depends on: Standard deviation in the population σ Total sample size N The precision is

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Precision of the Estimated Treatment Effect Precision in complex (clustered sample) designs depends on: The (total) standard deviation σ T Sample size at each level of sampling (e.g., m clusters, n individuals per cluster) Intraclass correlation structure It is a little harder to compute than in simple designs, but important because it helps you see what matters in design

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Intraclass Correlations in Two-level Designs In two-level designs the intraclass correlation structure is determined by a single intraclass correlation This intraclass correlation is the proportion of the total variance that is between schools (clusters)

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Precision in Two-level Hierarchical Design With No Covariates The standard error of the treatment effect is SE decreases as m (number of schools) increases SE deceases as n increases, but only up to point SE increases as ρ increases

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Statistical Power Power in simple (simple random sample) designs depends on: Significance level Effect size Sample size Look power up in a table for sample size and effect size

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Fragment of Cohen’s Table d n … … … … …

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Computing Statistical Power Power in complex (clustered sample) designs depends on: Significance level Effect size δ Sample size at each level of sampling (e.g., m clusters, n individuals per cluster) Intraclass correlation structure This makes it seem a lot harder to compute

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Computing Statistical Power Computing statistical power in complex designs is only a little harder than computing it for simple designs Compute operational effect size (incorporates sample design information) Δ T Look power up in a table for operational sample size and operational effect size This is the same table that you use for simple designs

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Power in Two-level Hierarchical Design With No Covariates Basic Idea: Operational Effect Size = (Effect Size) x (Design Effect) Δ T = δ x (Design Effect) For the two-level hierarchical design with no covariates Operational sample size is number of schools (clusters)

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Power in Two-level Hierarchical Design With No Covariates As m (number of schools) increases, power increases As effect size increases, power increases Other influences occur through the design effect As ρ increases the design effect (and power) decreases No matter how large n gets the maximum design effect is Thus power only increases up to some limit as n increases

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Two-level Hierarchical Design With Covariates (HLM Notation) Level 1 (individual level) Y ijk = β 0j + β 1j X ijk + ε ijk ε ~ N(0, σ AW 2 ) Level 2 (school Level) β 0j = π 00 + π 01 T j + π 02 W j + ξ 0j ξ ~ N(0, σ AS 2 ) β 1j = π 10 Note that the covariate effect β 1j = π 10 is a fixed effect If we code the treatment T j = ½ or - ½, then the parameters are identical to those in standard ANCOVA

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Precision in Two-level Hierarchical Design With Covariates The standard error of the treatment effect SE decreases as m increases SE deceases as n increases, but only up to point SE increases as ρ increases SE decreases as R W 2 and R S 2 increase

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Power in Two-level Hierarchical Design With Covariates Basic Idea: Operational Effect Size = (Effect Size) x (Design Effect) Δ T = δ x (Design Effect) For the two-level hierarchical design with covariates The covariates increase the design effect

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Power in Two-level Hierarchical Design With Covariates As m and effect size increase, power increases Other influences occur through the design effect As ρ increases the design effect (and power) decrease Now the maximum design effect as large n gets big is As the covariate-outcome correlations R W 2 and R S 2 increase the design effect (and power) increases

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Three-level Hierarchical Design Here there are three factors Treatment Schools (clusters) nested in treatments Classes (subclusters) nested in schools Suppose there are m schools (clusters) per treatment p classes (subclusters) per school (cluster) n students (individuals) per class (subcluster)

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Three-level Hierarchical Design With No Covariates The statistical model for the observation on the l th person in the k th class in the j th school in the i th treatment is Y ijkl = μ + α i + β j(i) + γ k(ij) + ε ijkl where μ is the grand mean, α i is the average effect of being in treatment i, β j(i) is the average effect of being in school j, in treatment i γ k(ij) is the average effect of being in class k in treatment i, in school j, ε ijkl is a residual

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Three-level Hierarchical Design With No Covariates (HLM Notation) Level 1 (individual level) Y ijkl = β 0jk + ε ijkl ε ~ N(0, σ W 2 ) Level 2 (classroom level) β 0jk = γ 0j + η 0jk η ~ N(0, σ C 2 ) Level 3 (school Level) γ 0j = π 00 + π 01 T j + ξ 0j ξ ~ N(0, σ S 2 ) If we code the treatment T j = ½ or - ½, then π 00 = μ, π 01 = α 1, ξ 0j = γ k(ij), η 0jk = β j(i)

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Three-level Hierarchical Design Intraclass Correlations In three-level designs there are two levels of clustering and two intraclass correlations At the school (cluster) level At the classroom (subcluster) level

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Precision in Three-level Hierarchical Design With No Covariates The standard error of the treatment effect SE decreases as m increases SE deceases as p and n increase, but only up to point SE increases as ρ S and ρ C increase

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Power in Three-level Hierarchical Design With No Covariates Basic Idea: Operational Effect Size = (Effect Size) x (Design Effect) Δ T = δ x (Design Effect) For the three-level hierarchical design with no covariates The operational sample size is the number of schools

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Power in Three-level Hierarchical Design With No Covariates As m and the effect size increase, power increases Other influences occur through the design effect As ρ S or ρ C increases the design effect decreases No matter how large n gets the maximum design effect is Thus power only increases up to some limit as n increases

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Three-level Hierarchical Design With Covariates (HLM Notation) Level 1 (individual level) Y ijkl = β 0jk + β 1jk X ijkl + ε ijkl ε ~ N(0, σ AW 2 ) Level 2 (classroom level) β 0jk = γ 00j + γ 01j Z jk + η 0jk η ~ N(0, σ AC 2 ) β 1jk = γ 10j Level 3 (school Level) γ 00j = π 00 + π 01 T j + π 02 W j + ξ 0j ξ ~ N(0, σ AS 2 ) γ 01j = π 01 γ 10j = π 10 The covariate effects β 1jk = γ 10j = π 10 and γ 01j = π 01 are fixed

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Precision in Three-level Hierarchical Design With Covariates SE decreases as m increases SE deceases as p and n increase, but only up to point SE increases as ρ increases SE decreases as R W 2, R C 2, and R S 2 increase

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Power in Three-level Hierarchical Design With Covariates Basic Idea: Operational Effect Size = (Effect Size) x (Design Effect) Δ T = δ x (Design Effect) For the three-level hierarchical design with covariates The operational sample size is the number of schools

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Power in Three-level Hierarchical Design With Covariates As m and the effect size increase, power increases Other influences occur through the design effect As ρ S or ρ C increase the design effect decreases No matter how large n gets the maximum design effect is Thus power only increases up to some limit as n increases

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Randomized Block Designs

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Two-level Randomized Block Design With No Covariates (HLM Notation) Level 1 (individual level) Y ijk = β 0j + β 1j T ijk + ε ijk ε ~ N(0, σ W 2 ) Level 2 (school Level) β 0j = π 00 + ξ 0j ξ 0j ~ N(0, σ S 2 ) β 1j = π 10 + ξ 1j ξ 1j ~ N(0, σ T x S 2 ) If we code the treatment T ijk = ½ or - ½, then the parameters are identical to those in standard ANOVA

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Randomized Block Designs In randomized block designs, as in hierarchical designs, the intraclass correlation has an impact on precision and power However, in randomized block designs designs there is also a parameter reflecting the degree of heterogeneity of treatment effects across schools We define this heterogeneity parameter ω S in terms of the amount of heterogeneity of treatment effects relative to the heterogeneity of school means Thus ω S = σ T x S 2 /σ S 2

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Precision in Two-level Randomized Block Design With No Covariates The standard error of the treatment effect SE decreases as m (number of schools) increases SE deceases as n and p increase, but only up to point SE increases as ρ increases SE increases as ω S = σ T x S 2 /σ S 2 increases

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Power in Two-level Randomized Block Design With No Covariates Basic Idea: Operational Effect Size = (Effect Size) x (Design Effect) Δ T = δ x (Design Effect) For the two-level hierarchical design with no covariates Operational sample size is number of schools (clusters)

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Precision in Two-level Randomized Block Design With Covariates The standard error of the treatment effect SE decreases as m increases SE deceases as n increases, but only up to point SE increases as ρ increases SE increases as ω S = σ T x S 2 /σ S 2 increases SE (generally) decreases as R W 2 and R S 2 increase

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Power in Two-level Randomized Block Design With Covariates Basic Idea: Operational Effect Size = (Effect Size) x (Design Effect) Δ T = δ x (Design Effect) For the two-level hierarchical design with covariates The covariates increase the design effect

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Three-level Randomized Block Designs

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Three-level Randomized Block Design With No Covariates Here there are three factors Treatment Schools (clusters) nested in treatments Classes (subclusters) nested in schools Suppose there are m schools (clusters) per treatment 2 p classes (subclusters) per school (cluster) n students (individuals) per class (subcluster)

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Three-level Randomized Block Design With No Covariates The statistical model for the observation on the l th person in the k th class in the i th treatment in the j th school is Y ijkl = μ +α i + β j + γ k + αβ ij + ε ijkl where μ is the grand mean, α i is the average effect of being in treatment i, β j is the average effect of being in school j, γ k is the effect of being in the k th class, αβ ij is the difference between the average effect of treatment i and the effect of that treatment in school j, ε ijkl is a residual

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Three-level Randomized Block Design With No Covariates (HLM Notation) Level 1 (individual level) Y ijkl = β 0jk + ε ijkl ε ~ N(0, σ W 2 ) Level 2 (classroom level) β 0jk = γ 00j + γ 01j T j + η 0jk η ~ N(0, σ C 2 ) Level 3 (school Level) γ 00j = π 00 + ξ 0j ξ oi ~ N(0, σ S 2 ) γ 01j = π 10 + ξ 1j ξ 1i ~ N(0, σ T x S 2 ) If we code the treatment T j = ½ or - ½, then π 00 = μ, π 10 = α 1, ξ 0j = β j, ξ 1j = αβ ij, η 0jk = γ k

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Three-level Randomized Block Design Intraclass Correlations In three-level designs there are two levels of clustering and two intraclass correlations At the school (cluster) level At the classroom (subcluster) level

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Three-level Randomized Block Design Heterogeneity Parameters In three-level designs, as in two-level randomized block designs, there is also a parameter reflecting the degree of heterogeneity of treatment effects across schools We define this parameter ω S in terms of the amount of heterogeneity of treatment effects relative to the heterogeneity of school means (just like in two-level designs) Thus ω S = σ T x S 2 /σ S 2

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Precision in Three-level Randomized Block Design With No Covariates The standard error of the treatment effect SE decreases as m increases SE deceases as p and n increase, but only up to point SE increases as ω S increases SE increases as ρ S and ρ C increase

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Power in Three-level Randomized Block Design With No Covariates Basic Idea: Operational Effect Size = (Effect Size) x (Design Effect) Δ T = δ x (Design Effect) For the three-level hierarchical design with no covariates The operational sample size is the number of schools

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Power in Three-level Randomized Block Design With No Covariates As m and the effect size increase, power increases Other influences occur through the design effect As ρ S or ρ C increases the design effect decreases No matter how large n gets the maximum design effect is Thus power only increases up to some limit as n increases

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Power in Three-level Randomized Block Design With Covariates SE decreases as m increases SE deceases as p and n increases, but only up to point SE increases as ρ and ω S increase SE decreases as R W 2, R C 2, and R S 2 increase

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Power in Three-level Randomized Block Design With Covariates Basic Idea: Operational Effect Size = (Effect Size) x (Design Effect) Δ T = δ x (Design Effect) For the three-level hierarchical design with covariates The operational sample size is the number of schools

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Power in Three-level Randomized Block Design With Covariates As m and the effect size increase, power increases Other influences occur through the design effect As ρ S or ρ C increases the design effect decreases No matter how large n gets the maximum design effect is Thus power only increases up to some limit as n increases

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What Unit Should Be Randomized? (Schools, Classrooms, or Students) Experiments cannot estimate the causal effect on any individual Experiments estimate average causal effects on the units that have been randomized If you randomize schools the (average) causal effects are effects on schools If you randomize classes, the (average) causal effects are on classes If you randomize individuals, the (average) causal effects estimated are on individuals

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What Unit Should Be Randomized? (Schools, Classrooms, or Students) Theoretical Considerations Decide what level you care about, then randomize at that level Randomization at lower levels may impact generalizability of the causal inference (and it is generally a lot more trouble) Suppose you randomize classrooms, should you also randomly assign students to classes? It depends: Are you interested in the average causal effect of treatment on naturally occurring classes or on randomly assembled ones?

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What Unit Should Be Randomized? (Schools, Classrooms, or Students) Relative power/precision of treatment effect Assign Schools (Hierarchical Design) Assign Classrooms (Randomized Block) Assign Students (Randomized Block)

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What Unit Should Be Randomized? (Schools, Classrooms, or Students) Precision of estimates or statistical power dictate assigning the lowest level possible But the individual (or even classroom) level will not always be feasible or even theoretically desirable

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Questions and Answers About Design

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1.Is it ok to match my schools (or classes) before I randomize to decrease variation? 2.I assigned treatments to schools and am not using classes in the analysis. Do I have to take them into account in the design? 3.I am assigning schools, and using every class in the school. Do I have to include classes as a nested factor? 4.My schools all come from two districts, but I am randomly assigning the schools. Do I have to take district into account some way?

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Questions and Answers About Design 1.I didn’t really sample the schools in my experiment (who does?). Do I still have to treat schools as random effects? 2.I didn’t really sample my schools, so what population can I generalize to anyway? 3. I am using a randomized block design with fixed effects. Do you really mean I can’t say anything about effects in schools that are not in the sample?

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Questions and Answers About Design 1.We randomly assigned, but our assignment was corrupted by treatment switchers. What do we do? 2.We randomly assigned, but our assignment was corrupted by attrition. What do we do? 3.We randomly assigned but got a big imbalance on characteristics we care about (gender, race, language, SES). What do we do? 4.We randomly assigned but when we looked at the pretest scores, we see that we got a big imbalance (a “bad randomization”). What do we do?

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Questions and Answers About Design 1.We care about treatment effects, but we really want to know about mechanism. How do we find out if implementation impacts treatment effects? 2.We want to know where (under what conditions) the treatment works. Can we analyze the relation between conditions and treatment effect to find this out? 3.We have a randomized block design and find heterogeneous treatment effects. What can we say about the main effect of treatment in the presence of interactions?

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Questions and Answers About Design 1.I prefer to use regression and I know that regression and ANOVA are equivalent. Why do I need all this ANOVA stuff to design and analyze experiments? 2.Don’t robust standard errors in regression solve all these problems? 3.I have heard of using “school fixed effects” to analyze a randomized block design. Is the a good alternative to ANOVA or HLM? 4.Can I use school fixed effects in a hierarchical design?

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Questions and Answers About Design 1.We want to use covariates to improve precision, but we find that they act somewhat differently in different groups (have different slopes). What do we do? 2.We get somewhat different variances in different groups. Should we use robust standard errors? 3.We get somewhat different answers with different analyses. What do we do?

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